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example of an extension that is not normal

Consider the extensionℚ⁢(23)/ℚℚ32ℚ\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}. The minimal polynomial for 2332\sqrt[3]{2} over ℚℚ\mathbb{Q} is x3-2superscriptx32x^{3}-2. This polynomialfactors in ℚ⁢(23)ℚ32\mathbb{Q}(\sqrt[3]{2}) as x3-2=(x-23)⁢(x2+x⁢23+43)superscriptx32x32superscriptx2x3234x^{3}-2=(x-\sqrt[3]{2})(x^{2}+x\sqrt[3]{2}+\sqrt[3]{4}). Let f⁢(x)=x2+x⁢23+43fxsuperscriptx2x3234f(x)=x^{2}+x\sqrt[3]{2}+\sqrt[3]{4}. Note that disc⁡(f⁢(x))=(23)2-4⁢43=43-4⁢43=-3⁢43<0discfxsuperscript322434344343340\operatorname{disc}(f(x))=(\sqrt[3]{2})^{2}-4\sqrt[3]{4}=\sqrt[3]{4}-4\sqrt[3]%
{4}=-3\sqrt[3]{4}<0. Thus, f⁢(x)fxf(x) has no real roots. Therefore, f⁢(x)fxf(x) has no roots in ℚ⁢(23)ℚ32\mathbb{Q}(\sqrt[3]{2}) since ℚ⁢(23)⊆ℝℚ32ℝ\mathbb{Q}(\sqrt[3]{2})\subseteq\mathbb{R}. Hence, x3-2superscriptx32x^{3}-2 has a root in ℚ⁢(23)ℚ32\mathbb{Q}(\sqrt[3]{2}) but does not split in ℚ⁢(23)ℚ32\mathbb{Q}(\sqrt[3]{2}). It follows that the extension ℚ⁢(23)/ℚℚ32ℚ\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q} is not normal.